I would like to finish this lecture, as well as discuss with this discussion of the so-called Levy-based Models. So, these are models which are based on Levy processes. The reason why one should introduce more complicated models than the Levy processes is very simple. This is because Levy processes are sometimes too simple to describe real data. For instance, in the context of financial data, there are so-called Stylized-facts of financial data. And these facts are posed by economists. These are something like requirements which should be included in your model, in order to realistically describe the financial data. Well, I would like to present a couple of models which are from this class. Let me start with the so-called Time-changed models. And these models are based on the idea of Stochastic time change. The idea is the following, we consider a process X T, usually a Levy process, and then change time here by another Stochastic process capital T of S. For instance, capital T can be also Levy process, but of course it's natural to consider sub-processes capital T of S, which are increasing, because time should increase. And therefore, this Levy process should be increasing, in other words it should be a subordinator. Finally, we get a process X T of S. This is a process with time S. Economical interpretation of this kind of model is the following one. Well, assume that we analyze the stock prices, and we construct a Logarithm of S T divided by S zero. This is returned from zero to T and Logarithm of this vector. Well, when the amount of transactions is high, then the picture is, let me say, more intensive. And when the amount of transactions is small, then it is more, let me say, flexible. And if you look at this picture, you may have the following idea. So, you might think that there is a process, for instance, a Brownian motion, and sometimes this process is made more narrow, sometimes more broader, and finally we get this picture. So, if you change time in the Levy process, then somehow it can be used for modeling the dynamics of this type. Okay, this idea is widely used in finance, and basically this idea means that Logarithm of S T divided by S zero is equal to Brownian motion with change time. And a natural candidate for capital T of S is the cumulative amount of transactions from zero to S. The idea of Stochastic time changed can be used in economics, but it also has some very good mathematical background due to the so called Monroe's Theorem. The theorem tells us that the class of subordinated Brownian motions, basically, coincides with a class of Semi-martingales. Well, this theorem is rather difficult. And in this theorem, Brownian motion and T of S can be dependent. And for practical analysis of this kind of models, is much more simpler to assume that they are independent. Because in this case, the Characteristic function of the process X T S, is equal to the superposition of the Laplace transform of the process T of S at the point minus psi of U. Where psi is the characteristic exponent of process X. Now, this formula allows to analyze this kind of processes, because one can derive a lot of facts from this formula. For instance, that if T and X are both Levy processes, and they are independent, then the superposition is in fact also Levy process. On the other side, one can calculate a Levy Triplet of the resulting process. So, namely this formula is a very important object in this theory. So, this is the first idea of Stochastic time change. The second idea, which I would like also to shortly discuss now, is the so-called Stochastic Volatility. If we return to the Black Scholes Model, which was considered on the previous lecture, you know that D of Logarithm S T in this model is equal to mu minus sigma squared denoted by 2 D T plus sigma D W T, where W is a Brownian motion. And here sigma is something similar to the Volatility parameter. The question which was raised by economists that Sigma cannot be considered as a constant. It also changes over time, and moreover, there is some economical evidence that this should change simultaneously with the changes in price. The mathematical idea how I can incorporate this type of idea, this type of economical wishes, is to change sigma by another random process, V of T, which should be at least non-negative. To say that this V of T is a Stochastic Volatility. And this idea gives rise for the so-called Stochastic volatility models. For instance, for this process V T, one can use the so called Cox–Ingersoll–Ross process, which can be defined V as a following Stochastic differential equations. D V T is equal to A minus B multiplied by V T D T, plus the constant C multiplied by square-root of V T D T. A, B and C are three positive constants. So, what we have here, is that the price is a Stochastic process, and also the volatility is also a Stochastic process. So, we described the price, we have two Stochastic processes. And here in this process, Sigma is nothing more than V T. So, V T here, and square-root of V T here. Okay, these are two ideas which are widely used in the theory of Levy processes. And I think that is a very good moment to stop our lectures. Here on this board, you see a lot of ideas which were exactly the topics of this course, like Levy processes, Time-change, Laplace transform, Characteristic functions, Stochastic differential equations, Volatility and so on. It was my big pleasure to give these lectures on Coursera, and I hope very much that these lectures will help you to understand more complicated theories, like theory of Quantitative Finance or Population Dynamics. I wish you good luck.
